Microeconomic Theory Homework Help

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*QUESTION 1 Let x be a random variable with n discrete outcomes x1,...,xn. The probability of each outcome depends on a privately chosen action by an agent. The agent can either choose L or H. If the agent chooses L, the probability vector for x is pL = (pL(1),..,pL(n)). If the agent chooses H, the probability vector for x is pH = (pH(1),..,pH(n)). Assume that the Monotone Likelihood Ratio Property (MLRP) holds strictly, that is, pH(i)/pL(i) is strictly increasing in i.

The agent's preferences can be represented by the separable utility function u(s,e) = s - c(e), where s is the payment by the principal to the agent and e is the agent's unobserved action (e = L or H). Assume c(H) > c(L) = 0. The principal's utility is v(x-s) = x - s. Thus, both the agent and the principal are risk neutral. However, the agent has no wealth, which constrains the principal's payment to be non-negative: si ≥ 0 for every outcome xi. a. Set up the program that implements H at the minium expected cost to the principal (accounting for the agent's limited liability constraint). b. Show that in the minimum cost contract, si = 0 for i = 1,..,n-1 and sn > 0.

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QUESTION 2.

Adam is thinking about buying fire insurance for his summer house. The probability that the house is going to catch fire is assessed to be one in ten (.10). This probability is exogenous and not affected by anything Adam does. If the house catches fire, the damage is going to be either 1000 or 2000 with probabilities depending on whether Adam has prepared himself for a fire or not (bought fire extinguisher, etc). Adam’s opportunity cost of preparing himself for a fire is c > 0; assume Adam’s level of preparation cannot be observed by the insurance company. If Adam is prepared, the probability that the damage from the fire is 2000 equals (.5). If Adam isn’t prepared, that same probability is (.7).

Adam’s utility function takes the form: u(m) − {cost of preparing for fire}. Here u is a strictly concave utility function over money m. a. Set up a program that identifies the second-best insurance plan in which Adam prepares himself for a fire and the insurance company makes zero profits. b. Solve for the second-best solution in part (a) in terms of payoffs to Adam in the three cases: no fire, fire and damage is 1000, fire and damage is 2000 (use notation S1, S2, S3 for Adam’s payoffs in these three cases). You will not be able to solve payoffs in closed form, but you should be able to provide a characterization and interpretation of the second-best contract.

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Suppose now that Adam has a third option: he can prepare for a fire in a manner that would entirely eliminate the chance of incurring the 2000 damage (at some higher cost C > c). Describe qualitatively the optimal insurance contract that implements this new level of preparedness in the most efficient manner? *QUESTION 3.

In the linear incentive model, assume the cost function of the agent is c(e, s) = ½(e + s)2 − s, where e is time put into working for the principal and s is time spent with family and friends. The negative sign in front of s comes from the fact that the agent values time spent with family and friends. The agent’s utility function is exponential with risk aversion coefficient r = 2. Output from work is x = e + θ, where θ is Normally distributed with mean zero and variance 4. a. Suppose the incentive coefficient α = .3. How much time will the agent spend on family and friends? b. Suppose the principal can set s = 0 by asking the agent to come to the office each day. What is the optimal choice of α in that case? c. Suppose the agent’s best alternative to working for the principal is spending time with family and friends. What will the principal have to pay the agent as a fixed salary (β) in part b?

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QUESTION 4.

This question is about optimal contracting in a principal-agent model with uniform distribution function. Let the production technology be described by x = e + θ. The agent chooses "effort" e privately (the principal cannot observe e). The principal observes x, which therefore can be used as a basis for contracting. Assume θ is distributed uniformly on [0,1]. The agent's utility function takes the form U(m,e) = u(m) - c(e), over money m and effort e. Here u is concave, c is convex, and both are increasing functions. The principal is risk neutral. a. What is the first-best contract (ie the contract when e can be observed and contracted on)? b. Show that the first-best outcome can be achieved even when e is not observed by the principal. That is, show that in the special case of a uniform distribution the principal-agent problem does not entail any welfare loss relative to the first-best.

Solution Question 1 (a) The program is the following:

The first constraint is the agent’s IC constraint, and the second is the limited liability constraint. (b) Let µ and νi be the Lagrange multipliers of the two constraints, respectively. Then the first-order conditions can be written as follows:

Notice that all multipliers are non-negative and a multiplier is zero only if its corresponding constraint is not binding

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Suppose that µ = 0. Then νi = 1 for all i which implies that si = 0 for all i, which cannot satisfy the agent’s IC constraint. Therefore, µ > 0. The strict MLRP implies that (pH(i) − pL(i))µ/pH(i) is strictly increasing in i, which further implies that νi is strictly decreasing in i. Therefore, νi > 0 and si = 0 for i = 1, 2, ..., n − 1. Finally, sn > 0 since otherwise the agent’s IC constraint cannot be satisfied.

Question 2 (a) Let pNF be the probability that there is no fire, p2e the probability that damage is 2000 when Adam exerts effort and p2ne the probability that damage is 2000 when Adam does not exert effort. Let x1, x2 and x3 be Adam’s consumption in these three cases. Here is the program:

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where W is Adam’s initial wealth. (b) If Adam’s IC constraint is not binding, x2 = x3 at optimality and the IC constraint is violated. If the company’s IR constraint is not binding, then all x’s are inifite. Therefore, both constriants are binding. Let λ > 0 and µ > 0 be Lagrange multipliers of the two constriants. Then the FOCs are

Therefore, u' (x2) < u' (x1) < u' (x3). Since u is strictly concave, u' is strictly decreasing, so x3 < x1 < x2. The same ordering is true for the S’s. Here is the intuition: we want to encourage Adam to exert the effort, so we need x2 > x3 to give him incentive. On the other hand, we want to insure him against the risk of fire, so we want to choose x1 in between.

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(c) Since the damage of 2000 does not occur in reality, we can make Adam’s payoff as low as possible in that case. Then we set S1 = S2 and check if the IC constraint is satisfied. If for some reason (such as limited liability) the IC constraint is not satisfied, then S1 < S2.

Question 3 (a) The agent maximizes

For α = 0.3, e = 0 and s = 1. (b) In this case, the agent chooses effort α, and the principal maximizes

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Question 4 (a) If e can be contracted on, the principal can write a contract to maximize total surplus, eliminate risk to the agent, and place the agent exactly at the minimum utility by choosing e and paying only if that level of e is observed. Note that with a risk averse agent and risk neutral principal, optimal risk sharing requires the principal to bear all risk and pay the agent a fixed amount. The principal maximizes

This condition together with the participation constraint gives a unique first-best effort e ∗ and associated payment s . (b) For simplicity, I assume c(e) ≥ 0, ∀e; the problem can be solved without this but it is slightly more tedious. Define s such that u(s) = min{u, u(s∗) − u' (s∗)}; by construction s < s∗. Consider the contract

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When the agent chooses e = e∗, then the agent’s utility is u; this is the first-best efficient solution and satisfies IR by construction. All that remains is to verify IC. If the agent chooses e > e∗, then she increases her disutility of effort and lowers her expected payout by placing positive probability on outcomes that pay s < s ∗, therefore she will not choose e > e∗. When e ≤ e∗ − 1, the agent’s expected utility is at most u − c(e) ≤ u so the agent will not prefer that either. For e = e ∗ − δ, δ ∈ (0, 1), the agent’s expected utility is

where the first inequality follows from the convexity of c(e). Thus the agent will not prefer e in that region, meaning IC is satisfied and the first-best is achieved.